Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

# Characterization of Relationships Between the Domains of Two Linear Matrix-Valued Functions with Applications

Version 1 : Received: 2 May 2020 / Approved: 3 May 2020 / Online: 3 May 2020 (06:23:35 CEST)

How to cite: Tian, Y. Characterization of Relationships Between the Domains of Two Linear Matrix-Valued Functions with Applications. Preprints 2020, 2020050025. https://doi.org/10.20944/preprints202005.0025.v1 Tian, Y. Characterization of Relationships Between the Domains of Two Linear Matrix-Valued Functions with Applications. Preprints 2020, 2020050025. https://doi.org/10.20944/preprints202005.0025.v1

## Abstract

One of the typical forms of linear matrix expressions (linear matrix-valued functions) is given by $A + B_1X_1C_1 + \cdots + B_kX_kC_k$, where $X_1, \ldots, X_k$ are independent variable matrices of appropriate sizes, which include almost all matrices with unknown entries as its special cases. The domain of the matrix expression is defined to be all possible values of the matrix expressions with respect to $X_1, \ldots, X_k$. I this article, we approach some problems on the relationships between the domains of two linear matrix expressions by means of the block matrix method (BMM), the matrix rank method (MRM), and the matrix equation method (MEM). As application, we discuss some topics on the relationships among general solutions of some linear matrix equations and their reduced equations.

## Keywords

linear matrix expression; domain; matrix equation; general solution; generalized inverse

## Subject

Computer Science and Mathematics, Mathematics